• Previous Article
    Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity
  • DCDS Home
  • This Issue
  • Next Article
    Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations
November  2014, 34(11): 4617-4645. doi: 10.3934/dcds.2014.34.4617

Blow-up set for a superlinear heat equation and pointedness of the initial data

1. 

Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka 560-8531, Japan

Received  October 2013 Revised  March 2014 Published  May 2014

We study the blow-up problem for a superlinear heat equation \begin{equation} \label{eq:P} \tag{P} \left\{ \begin{array}{ll} \partial_t u = \epsilon \Delta u + f(u),                      x\in\Omega, \,\,\, t>0, \\ u(x,t)=0,                                       x\in\partial\Omega, \,\,\, t>0, \\ u(x,0)=\varphi(x)\ge 0\, (\not\equiv 0),       x\in\Omega, \end{array} \right. \end{equation} where $\partial_t=\partial/\partial t$, $\epsilon>0$ is a sufficiently small constant, $N\ge 1$, $\Omega\subset {\bf R}^N$ is a domain, $\varphi\in C^2(\Omega)\cap C(\overline{\Omega})$ is a nonnegative bounded function, and $f$ is a positive convex function in $(0,\infty)$. In [10], the author of this paper and Ishige characterized the location of the blow-up set for problem (p) with $f(u)=u^p$ ($p>1$) with the aid of the invariance of the equation under some scale transformation for the solution, which played an important role in their argument. However, due to the lack of such scale invariance for problem (p), we can not apply their argument directly to problem (p). In this paper we introduce a new transformation for the solution of problem (p), which is a generalization of the scale transformation introduced in [10], and generalize the argument of [10]. In particular, we show the relationship between the blow-up set for problem (p) and pointedness of the initial function under suitable assumptions on $f$.
Citation: Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617
References:
[1]

X. Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations,, J. Differential Equations, 78 (1989), 160.  doi: 10.1016/0022-0396(89)90081-8.  Google Scholar

[2]

T. Cheng and G. F. Zheng, Some blow-up problems for a semilinear parabolic equation with a potential,, J. Differential Equations, 244 (2008), 766.  doi: 10.1016/j.jde.2007.11.004.  Google Scholar

[3]

C. Cortazar, M. Elgueta and J. D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential,, J. Math. Anal. Appl., 335 (2007), 418.  doi: 10.1016/j.jmaa.2007.01.079.  Google Scholar

[4]

A. Friedman and A. A. Lacey, The blow-up time for solutions of nonlinear heat equations with small diffusion,, SIAM J. Math. Anal., 18 (1987), 711.  doi: 10.1137/0518054.  Google Scholar

[5]

A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425.  doi: 10.1512/iumj.1985.34.34025.  Google Scholar

[6]

Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion,, Differential Integral Equations, 25 (2012), 759.   Google Scholar

[7]

Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation with small diffusion,, J. Differential Equations, 249 (2010), 1056.  doi: 10.1016/j.jde.2010.03.028.  Google Scholar

[8]

Y. Fujishima and K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on $R^N$,, J. Differential Equations, 250 (2011), 2508.  doi: 10.1016/j.jde.2010.12.008.  Google Scholar

[9]

Y. Fujishima and K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on $R^N$. II,, J. Differential Equations, 252 (2012), 1835.  doi: 10.1016/j.jde.2011.08.040.  Google Scholar

[10]

Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation and pointedness of the initial data,, Indiana Univ. Math. J., 61 (2012), 627.  doi: 10.1512/iumj.2012.61.4596.  Google Scholar

[11]

Y. Fujishima and K. Ishige, Blow-up set for type I blowing up solutions for a semilinear heat equation,, Ann. Inst. H. Poincaré Anal., 31 (2014), 231.  doi: 10.1016/j.anihpc.2013.03.001.  Google Scholar

[12]

Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations,, Comm. Pure Appl. Math., 42 (1989), 845.  doi: 10.1002/cpa.3160420607.  Google Scholar

[13]

K. Ishige, Blow-up time and blow-up set of the solutions for semilinear heat equations with large diffusion,, Adv. Differential Equations, 7 (2002), 1003.   Google Scholar

[14]

K. Ishige and N. Mizoguchi, Location of blow-up set for a semilinear parabolic equation with large diffusion,, Math. Ann., 327 (2003), 487.  doi: 10.1007/s00208-003-0463-4.  Google Scholar

[15]

K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion,, J. Differential Equations, 212 (2005), 114.  doi: 10.1016/j.jde.2004.10.021.  Google Scholar

[16]

N. Mizoguchi and E. Yanagida, Life span of solutions with large initial data in a semilinear parabolic equation,, Indiana Univ. Math. J., 50 (2001), 591.  doi: 10.1512/iumj.2001.50.1905.  Google Scholar

[17]

N. Mizoguchi and E. Yanagida, Life span of solutions for a semilinear parabolic problem with small diffusion,, J. Math. Anal. Appl., 261 (2001), 350.  doi: 10.1006/jmaa.2001.7530.  Google Scholar

[18]

P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States,, Birkhäuser Advanced Texts: Basler Lehrbücher, (2007).  doi: 10.1007/978-3-7643-8442-5.  Google Scholar

[19]

S. Sato, Life span of solutions with large initial data for a superlinear heat equation,, J. Math. Anal. Appl. 343 (2008), 343 (2008), 1061.  doi: 10.1016/j.jmaa.2008.02.018.  Google Scholar

[20]

J. J. L. Velázquez, Higher-dimensional blow-up for semilinear parabolic equations,, Comm. Partial Differential Equations, 17 (1992), 1567.  doi: 10.1080/03605309208820896.  Google Scholar

[21]

J. J. L. Velázquez, Estimates on the $(n-1)$-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation,, Indiana Univ. Math. J., 42 (1993), 445.  doi: 10.1512/iumj.1993.42.42021.  Google Scholar

[22]

F. B. Weissler, Single point blow-up for a semilinear initial value problem,, J. Differential Equations 55 (1984), 55 (1984), 204.  doi: 10.1016/0022-0396(84)90081-0.  Google Scholar

[23]

H. Yagisita, Blow-up profile of a solution for a nonlinear heat equation with small diffusion,, J. Math. Soc. Japan, 56 (2004), 993.  doi: 10.2969/jmsj/1190905445.  Google Scholar

[24]

H. Yagisita, Variable instability of a constant blow-up solution in a nonlinear heat equation,, J. Math. Soc. Japan, 56 (2004), 1007.  doi: 10.2969/jmsj/1190905446.  Google Scholar

[25]

H. Zaag, On the regularity of the blow-up set for semilinear heat equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 505.  doi: 10.1016/S0294-1449(01)00088-9.  Google Scholar

[26]

H. Zaag, One-dimensional behavior of singular $N$-dimensional solutions of semilinear heat equations,, Comm. Math. Phys., 225 (2002), 523.  doi: 10.1007/s002200100589.  Google Scholar

[27]

H. Zaag, Regularity of the blow-up set and singular behavior for semilinear heat equations,, Mathematics mathematics education (Bethlehem, (2000), 337.   Google Scholar

[28]

H. Zaag, Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation,, Duke Math. J., 133 (2006), 499.  doi: 10.1215/S0012-7094-06-13333-1.  Google Scholar

show all references

References:
[1]

X. Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations,, J. Differential Equations, 78 (1989), 160.  doi: 10.1016/0022-0396(89)90081-8.  Google Scholar

[2]

T. Cheng and G. F. Zheng, Some blow-up problems for a semilinear parabolic equation with a potential,, J. Differential Equations, 244 (2008), 766.  doi: 10.1016/j.jde.2007.11.004.  Google Scholar

[3]

C. Cortazar, M. Elgueta and J. D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential,, J. Math. Anal. Appl., 335 (2007), 418.  doi: 10.1016/j.jmaa.2007.01.079.  Google Scholar

[4]

A. Friedman and A. A. Lacey, The blow-up time for solutions of nonlinear heat equations with small diffusion,, SIAM J. Math. Anal., 18 (1987), 711.  doi: 10.1137/0518054.  Google Scholar

[5]

A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425.  doi: 10.1512/iumj.1985.34.34025.  Google Scholar

[6]

Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion,, Differential Integral Equations, 25 (2012), 759.   Google Scholar

[7]

Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation with small diffusion,, J. Differential Equations, 249 (2010), 1056.  doi: 10.1016/j.jde.2010.03.028.  Google Scholar

[8]

Y. Fujishima and K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on $R^N$,, J. Differential Equations, 250 (2011), 2508.  doi: 10.1016/j.jde.2010.12.008.  Google Scholar

[9]

Y. Fujishima and K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on $R^N$. II,, J. Differential Equations, 252 (2012), 1835.  doi: 10.1016/j.jde.2011.08.040.  Google Scholar

[10]

Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation and pointedness of the initial data,, Indiana Univ. Math. J., 61 (2012), 627.  doi: 10.1512/iumj.2012.61.4596.  Google Scholar

[11]

Y. Fujishima and K. Ishige, Blow-up set for type I blowing up solutions for a semilinear heat equation,, Ann. Inst. H. Poincaré Anal., 31 (2014), 231.  doi: 10.1016/j.anihpc.2013.03.001.  Google Scholar

[12]

Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations,, Comm. Pure Appl. Math., 42 (1989), 845.  doi: 10.1002/cpa.3160420607.  Google Scholar

[13]

K. Ishige, Blow-up time and blow-up set of the solutions for semilinear heat equations with large diffusion,, Adv. Differential Equations, 7 (2002), 1003.   Google Scholar

[14]

K. Ishige and N. Mizoguchi, Location of blow-up set for a semilinear parabolic equation with large diffusion,, Math. Ann., 327 (2003), 487.  doi: 10.1007/s00208-003-0463-4.  Google Scholar

[15]

K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion,, J. Differential Equations, 212 (2005), 114.  doi: 10.1016/j.jde.2004.10.021.  Google Scholar

[16]

N. Mizoguchi and E. Yanagida, Life span of solutions with large initial data in a semilinear parabolic equation,, Indiana Univ. Math. J., 50 (2001), 591.  doi: 10.1512/iumj.2001.50.1905.  Google Scholar

[17]

N. Mizoguchi and E. Yanagida, Life span of solutions for a semilinear parabolic problem with small diffusion,, J. Math. Anal. Appl., 261 (2001), 350.  doi: 10.1006/jmaa.2001.7530.  Google Scholar

[18]

P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States,, Birkhäuser Advanced Texts: Basler Lehrbücher, (2007).  doi: 10.1007/978-3-7643-8442-5.  Google Scholar

[19]

S. Sato, Life span of solutions with large initial data for a superlinear heat equation,, J. Math. Anal. Appl. 343 (2008), 343 (2008), 1061.  doi: 10.1016/j.jmaa.2008.02.018.  Google Scholar

[20]

J. J. L. Velázquez, Higher-dimensional blow-up for semilinear parabolic equations,, Comm. Partial Differential Equations, 17 (1992), 1567.  doi: 10.1080/03605309208820896.  Google Scholar

[21]

J. J. L. Velázquez, Estimates on the $(n-1)$-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation,, Indiana Univ. Math. J., 42 (1993), 445.  doi: 10.1512/iumj.1993.42.42021.  Google Scholar

[22]

F. B. Weissler, Single point blow-up for a semilinear initial value problem,, J. Differential Equations 55 (1984), 55 (1984), 204.  doi: 10.1016/0022-0396(84)90081-0.  Google Scholar

[23]

H. Yagisita, Blow-up profile of a solution for a nonlinear heat equation with small diffusion,, J. Math. Soc. Japan, 56 (2004), 993.  doi: 10.2969/jmsj/1190905445.  Google Scholar

[24]

H. Yagisita, Variable instability of a constant blow-up solution in a nonlinear heat equation,, J. Math. Soc. Japan, 56 (2004), 1007.  doi: 10.2969/jmsj/1190905446.  Google Scholar

[25]

H. Zaag, On the regularity of the blow-up set for semilinear heat equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 505.  doi: 10.1016/S0294-1449(01)00088-9.  Google Scholar

[26]

H. Zaag, One-dimensional behavior of singular $N$-dimensional solutions of semilinear heat equations,, Comm. Math. Phys., 225 (2002), 523.  doi: 10.1007/s002200100589.  Google Scholar

[27]

H. Zaag, Regularity of the blow-up set and singular behavior for semilinear heat equations,, Mathematics mathematics education (Bethlehem, (2000), 337.   Google Scholar

[28]

H. Zaag, Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation,, Duke Math. J., 133 (2006), 499.  doi: 10.1215/S0012-7094-06-13333-1.  Google Scholar

[1]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[2]

Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194

[3]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243

[4]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[5]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[6]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

[7]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[8]

Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405

[9]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

[10]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

[11]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[12]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[13]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[14]

Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020401

[15]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151

[16]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[17]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205

[18]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453

[19]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[20]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (42)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]